Question

The one-to-one functions g and h are defined as follows. g=((-5, 2),( -3, 8), (-1, - 8), (8, 9))

h(x)=3x+2
Find the folowing:
g^-1 (8)=?
h^-1 (x)=?
(h^-1 \circ h)(-3)=?

Answer 1

Answer:
`g^(-1) (8)=-3`

`h^(-1)(x)=(x-2)/(3)`

`(h^(-1) \circ h)(-3)=-3`.

Explanation:

Given: The one-to-one functions g and h are defined as follows.

g=((-5, 2),( -3, 8), (-1, - 8), (8, 9)) [here each x= input values , y= output values in (x,y)]

h(x)=3x+2

To find:
`g^(-1) (8)`

As for 8 is a image of -3 ( from point ( -3, 8)).

So,
`g^(-1) (8)=-3`

To find :
`h^(-1) (x)`

Let
`y = h(x)=3x+2` (i)

Then,

`y=3x+2\Rightarrow\ 3x= y-2\\\\\Rightarrow\ x=(y-2)/(3)`

Switch
`y` with
`x` and
`x` with
`h^(-1) (x)`, we get

`h^(-1)(x)=(x-2)/(3)` (ii)

To find :
`(h^(-1) \circ h)(-3)`

Consider
`(h^(-1) \circ h)(-3)=h^(-1) (h(-3))`

`=h^(-1)(3(-3)+2)\ \ (using\ (i))\\\\=h^(-1)(-9+2)\\\\=h^(-1)(-7)\\\\= (-7-2)/(3)\ \ (using\ (ii))\\\\=(-9)/(3)=-3`

So,
`(h^(-1) \circ h)(-3)=-3`.